How long have you had these phones? The batteries may be coming to end of life and need replacing, they don't last forever. As a quick check you can try substituting the rechargeable batteries in one of your extension phones with normal alkalines but DO NOT PLACE IN THE CHARGING CRADLE. If the phones then work correctly you can fit new rechargeable batteries.
if your main phone base unit is working connected direct to hub - making/receiving calls - then digital voice is working. the problem now is that your phone base unit is not 'talking' to the extension phones
@Lila did you test calling each set from the other handsets - and could you hear all other handsets on each one?
Coincidences do happen and when changing one thing and something else changes for whatever reason it is human nature to blame the known change regardless. Another user on here found that his WiFi on his SH2 was terrible after switching to FTTP and DV - he eventually traced the problem to the WiFi aerial on his desktop PC not being connected to the PC nothing to do with the actual change by BT.
The symptoms you describe are very difficult for anyone to explain being related to moving to DV (at least at the moment) and seem much more likely to be within the BT3510 system - hence all the questions about that.
“I find it baffling that BT can't even help and I have to rely on members of the public”.
The reason for that is that you don’t deploy your more experienced and capable staff on the frontline helpdesk. A lot of the helpdesk traffic is trivial, (no offense meant to anyone), so you put your newbies on there and deploy the experienced staff on the more difficult backroom stuff.
A lot of the regulars on here are retired telecoms and IT techs etc., so strangely enough, you do tend to get the benefit of more experience on here.
This is absolutely nothing whatsoever to do with Digital Voice.
Very true but just offering the OP an explanation for her last statement.
So you agree then that BT value their customers so little that they can't be bothered to provide a help service? There is no point pretending to provide a service if they are staffed with 'newbies' who cannot advise.